Eigenvalue estimate for the p$p$‐Laplace operator on a connected finite graph

Abstract In this paper, we consider eigenvalue estimate for the ‐Laplace operator on a connected finite graph with the condition for and . We first establish elliptic gradient estimates for solutions to the eigenvalue equation. Then we establish a lower bound estimate for the first nonzero eigenvalue of , which is a generalization not only of the eigenvalue estimate for the manifold setting, but also of the eigenvalue estimate for the ‐Laplace operator on the graph with the condition.